The finite element method is applied to wave propagation and advection problems often used to develop models for numerical weather prediction. For these problems the error analysis must distinguish the error in representing the initial and final states from the error in evolution, for instance the error in wave speed. The analysis indicates that finite element methods can only compete with explicit finite difference schemes if "super-convergence" can be obtained. For the Galerkin method this means using splines on a regular or almost regular mesh. This prediction is confirmed in actual tests. Because the finite element approximation provides a complete field rather than gridpoint values alone, the analysis based on it is more meaningful than a gridpoint analysis, there is no "aliasing". The linear spline Galerkin method is then applied to test problems. The high accuracy for given resolution has to be balanced against the extra work involved. For linear problems the results agree with theory. The method is equivalent to an explicit fourth order method with 1.6 times the linear resolution; the efficiency is comparable to explicit finite difference schemes. The nonlinear problems solved include the shallow Water equations in a channel and on a sphere. The finite element algorithm is only competitive if the nonlinear terms involving derivatives are evaluated in two stages. If this is done the integration on a sphere is particularly successful. Artificial viscosity is necessary for realistic problems, the accuracy of the finite element method allows more refined forms of nonlinear viscosity to be used successfully. In general the spline Galerkin method is competitive with explicit finite difference schemes. For nonlinear problems it may give a fundamental improvement, as the spectral method does. However, extreme care has to be taken to get enough accuracy to balance the extra work.